In an accelerator when the bunch length becomes comparable to a
characteristic distance
s0, one which depends on the radius and the
conductivity of the beam tube and in typical structures is on the
order of tens of microns, the usual formulas for the resistive wall
wakefield do not apply. In this report we derive the short-range
resistive wall wakefields of an ultra-relativistic point particle in a
metallic. cylindrical tube, both for a model in which the wall
conductivity is taken to be independent of frequency and for one in
which a frequency dependence is included. On this scale the wakefield
is found to be dominated by a damped, high-frequency resonator
component. For the case of constant conductivity the resonant
frequency is given by
omega =
sqrt(3)c/s0
and the
Q-factor equals
sqrt(3)/2.
We provide a physical model to explain these results. For the case of
a frequency dependent conductivity the resonator parameters depend
also on the relaxation time of the metal tau. For
c tau/s0 >~ 0.5 the
frequency
omega is approximately equal to
sqrt(2 omegapc/b), with
omegap the plasma frequency of the free electrons
in the metal and b the tube radius, and the
1/e damping time becomes 4tau. Finally. we calculate
the wakefield and loss factor of a short Gaussian bunch.